Tokamak stability

I ntroduction to Nuclear Fusion

Prof. Dr. Yong-Su Na

2

Tokamak stability

3

- Considering plasma states which are not in perfect

thermodynamic equilibrium (no exact Maxwellian distribution, e.g. non-uniform density), even though they represent

equilibrium states in the sense that the force balance is equal to 0 and a stationary solution exists, means their entropy is not at the maximum possible and hence free energy appears

available which can excite perturbations to grow:

unstable equilibrium state

- The gradients of plasma current magnitude and pressure are the destabilising forces in connection with the bad magnetic field

curvature: The ratio of these two free energies turns out to be β_p

Tokamak Stability

4

• Definition of Stability

- Assuming all quantities of interest linearised about their equilibrium values.

 i t i t t

i _{r i r i}

e e

r Q e

r

Q





(  )

(  )

1

~ /

0

1

Q 

) Q ,

~ ( )

( )

,

( r t Q

r Q

r t

Q     

Im ω > 0 (ωi > 0): exponential instability Im ω ≤ 0 (ω_i ≤ 0): exponential stability

Stability

metastable

linearly unstable non-linearly unstable stable

i

r

i 



  

t

e

r Q t

r

Q ~ ( , )  ( )

1 1





5

• The Energy Principle

- Representing the most efficient and often the most intuitive method of determining plasma stability.

2

  0

K

 W



 0

 W

V S

F W W

W

W

  



 



 _

S S dS n n p B

W [[ /2 ]]

2 1

0 2 2









 V V

B r d W

0 2

ˆ1

2 1









































 _{  }

F P J Q p p

Q r d

W ^{* 2 *}

0 2

) (

) 2 (

1

    

 





- If the minimum value of potential energy is positive for all displacements, the system is stable.

- It it is negative for any displacement, the system is unstable.

Tokamak Stability

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Energy required to compress the plasma: main source of potential energy for the sound wave

Energy necessary to compress the magnetic field: major potential energy contribution to the compressional Alfvén wave

Energy required to bend magnetic field lines: dominant potential energy contribution to the shear Alfvén wave

destabilising

current-driven (kinks) modes (+ or -) Pressure-driven modes (+ or -)

stabilising













































 ^ _{     }

F P Q B p p J b Q

r d

W    



 2 2( )( ) ( )

2

1 _*

||

2 * 2

0 2

0 2

















 

 

• The Intuitive Form of δW

Tokamak Stability

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pressure driven instabilities (Rayleigh-Taylor or

interchange instability)

• Instabilities

- Two sources of free energy available:

plasma current

pressure gradient of a plasma

current driven instabilities (kink mode or sausage)

http://www.hosecouplingsuk.com/fabrication/fire-hose-assembly.htm

Tokamak Stability

x

x

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Flux conservation Topology unchanged

Reconnection of field lines Topology changed

•

- current driven (kink) instabilities internal modes

external modes

- pressure driven instabilities interchange modes ballooning modes

- current+pressure driven: Edge Localised Modes (ELMs) - vertical instability

•

- current driven instabilities tearing modes

neoclassical tearing modes (NTMs) - nonlinear modes

sawtooth disruption

•

Tokamak Stability

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Ideal MHD instabilities in a Tokamak

- fast growth (microseconds)

- the possible extension over the entire plasma

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• The most Virulent Instabilities

Ideal MHD Instabilities

• Kink modes

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q-profile

3

0 0.5 0.95

Normalised radius 1

5

2

Advanced scenario Baseline scenario

q95

4 Hybrid scenario

- Causing a contortion of the helical plasma column - Driven by the radial gradient of the toroidal current - External kind modes:

Fastest and most dangerous Arising mainly when q_a < 2

Ideal MHD Instabilities

http://www.maysville-online.com/news/local/tollesboro-home-destroyed-in-fire/article_a5e0eb4e-235b-5c7d-afee-74bf98c4e738.html http://www.bbc.co.uk/bitesize/higher/physics/radiation/waves/revision/1/

m: poloidal mode number m = 1

m = 2

m = 3

- Stabilising effect by the conducting wall and strong toroidal magnetic field

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Kruskal-Shafranov criterion:

stability condition for external kink mode for the worst case

 1 q

Determining plasma current limit

set by kink instabilities → safety factor

p P

a

I

B a

I R

aB B

R

q aB







 ^





/ 2

stabilising

destabilising

] MA [

/ 5

/

2 a

B

R

a

B R

I

 

 

2 1

0

/

0 0





 R I a

aB B

R q aB

KS p

a

 





Imposing an important constraint on tokamak operation:

toroidal current upper limit: Kruskal-Shafranov current (I < I_KS)

Ideal MHD Instabilities

• Kink modes

- A toroidally confined plasma sees ‘bad’ convex curvature of the helical magnetic field lines on the outboard side of the torus.

• Interchange modes

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x

x

http://blog.naver.com/PostView.nhn?blogId=ray0620&logNo=150112423635&parentCategoryNo=1&viewDate=&currentPage=1&listtype=0 http://en.wikipedia.org/wiki/File:St_Louis_Gateway_Arch.jpg

Ideal MHD Instabilities

F. F. Chen, “An Indispensable Truth”, Springer (2011)

• Interchange modes

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- A toroidally confined plasma sees ‘bad’ convex curvature of the helical magnetic field lines on the outboard side of the torus.

- The average curvature of B-field lines over a full poloidal rotation is ‘good’ for windings with a rotational transform ι ≤ 2π, i.e., q ≥ 1.

- Interchange perturbations do not grow in normal tokamaks if q ≥ 1.

Ideal MHD Instabilities

Unstable Stable

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• Ballooning modes

Ideal MHD Instabilities

- locally grow in the outboard bad curvature region: ballooning modes - A high local pressure gradient is responsible for driving the

ballooning instability.

- Can be suppressed almost everywhere in the plasma by establishing appropriate pressure profiles and appropriate magnetic field line

windings.

J.P. Freidberg, “Ideal Magneto-Hydro-Dynamics”, lecture note

https://kr.fotolia.com/tag/%ED%92%8D%EC%84%A0%EB%B6%88%EA%B8%B0

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• Edge Localised Modes (ELMs)

- current driven (peeling mode) and pressure driven (ballooning mode) combined instability

Ideal MHD Instabilities

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• Edge Localised Modes (ELMs)

Ideal MHD Instabilities

- current driven (peeling mode) and pressure driven (ballooning mode) combined instability

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A. Critical ∇p in H-mode barrier region reached

→ short unstable phase (ELM event)

B. Energy and particle loss reduces gradients.

C. Gradients build up during reheat/refuelling phase.

• Edge Localised Modes (ELMs)

Ideal MHD Instabilities

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LCFS

A. Critical ∇p in H-mode barrier region reached

→ short unstable phase (ELM event)

B. Energy and particle loss reduces gradients.

C. Gradients build up during reheat/refuelling phase.

• Edge Localised Modes (ELMs)

Ideal MHD Instabilities

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t = 2650 _A^-1 t = 2700 _A^{-1 t = 2890} _A^-1 - Non-linear MHD simulations with JOREK

Huysmans, Czarny, NF 47 659 (2007)

Evolution of ballooning mode

• Edge Localised Modes (ELMs)

Ideal MHD Instabilities

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- Non-linear MHD simulations with JOREK

• Edge Localised Modes (ELMs)

Ideal MHD Instabilities

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- Standard ELM dynamics in the KSTAR visualized by ECEI

1 2 3

100 ms 200 ms 0 ms

LCFS

(1) Initial Growth

4

400 ms

LCFS

(2) Saturation

G.S. Yun et al., PRL (2011)

• Edge Localised Modes (ELMs)

Ideal MHD Instabilities

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(3) ELM crash

5 ms 10 ms 15ms 25ms

0 ms

z (cm)

R (cm) R (cm)

LCFS

Particles/heat transport through the finger

Filaments elongate poloidally

A narrow finger- like structure

develops

- Standard ELM dynamics in the KSTAR visualized by ECEI

• Edge Localised Modes (ELMs)

Ideal MHD Instabilities

G.S. Yun et al., PRL (2011)

• Edge Localised Modes (ELMs)

Ideal MHD Instabilities

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Courtesy of Y. M. Jeon (NFRI)

- Full suppression by 3D magnetic perturbation

• Vertical Instability

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X I_p

Ideal MHD Instabilities

- Macroscopic vertical motion of the plasma towards the wall

J.P. Freidberg, “Ideal Magneto-Hydro-Dynamics”, lecture note

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Ideal MHD Instabilities

• Vertical Instability

J.P. Freidberg, “Ideal Magneto-Hydro-Dynamics”, lecture note

R

p

B

I  



B 

27

Ideal MHD Instabilities

• Vertical Instability

J.P. Freidberg, “Ideal Magneto-Hydro-Dynamics”, lecture note

R

p

B

I  



B 

KSTAR & ITER 28

- For a circular cross sections a moderate shaping of the vertical field should provide stability.

- For noncircular tokamaks, vertical instabilities produce important limitations on the maximum achievable elongations.

- Even moderate elongations require a conducting wall or a feedback system for vertical stability.

Ideal MHD Instabilities

• Vertical Instability

J.P. Freidberg, “Ideal Magneto-Hydro-Dynamics”, lecture note